Subject:
         Re:: Leap Week Calendar Proposal by W.Z.
   Date:
         Sun, 02 Mar 1997 13:28:43 -0800
   From:
         Simon Cassidy <scassidy@earthlink.net>
     To:
         CALNDR-L@ECUVM.CIS.ECU.EDU

Walter Ziobro wrote:
>         Under the system described here [leap-week], both the Greg.
>         Calendar and the Leap-Week Calendar have 146,097 days in
>         400 years.  While this is quite accurate, it is not quite
>         perfect, being too long by one day in about 3,320 years.

Simon Cassidy responds:
   Such statements as these are false for reasons I have given many
times already during this discussion. But, here I go again (for
Walter's sake at the least).
   The average Gregorian calendar year is 365.2425 days. The accuracy
of the Gregorian calendar must be judged against the average interval
between Vernal Equinoxes, since the Gregorian reform was intended to
prevent the drift of the astronomical Vernal Equinox away from the
Nicene "ecclesiastical Vernal Equinox" date (March 21st.).
   The average interval between Vernal Equinoxes is 365.2424 days (to
the nearest ten-thousandth of a day) and will probably remain so for
the next few millenia (it is increasing slowly towards a maximum
between 365.2424 and 365.2425 calendar days).
   See "Astronomical Algorithms" by Jean Meeus, 1991, for the sort of
formulae which must be used, to arrive at the true value of "the mean
interval between Vernal Equinoxes" as a function of time. I am also
working on a contribution to this discussion, which will use Meeus'
formulae for the instant of Vernal Equinox (given in my previous
message entitled "Re: Exigius 31/128 source not handy") and adapt
them for use by anyone in a spreadsheet program, so as to give the
real length of the Vernal Equinox year for any era from about 1000 B.C.
to about 3000 A.D.
   In the longer term the length of the year (however defined)
measured in real calendar (Universal) days, is not mechanistically
determinable since the future of the Earth's rotation rate is
dependent on human activities (i.e. it is subject to human free-will).
   Thus, in its own terms, the Gregorian calendar is currently too long
by one day in about 8,000 years and will actually get even more
accurate over the next millenia.

Walter Ziobro continued:
>         It has been proposed to rectify this by dropping a leap-
>         day in the Gregorian Calendar once every 4,000 years.
>         This would make the Gregorian Calendar accurate to within
>         one day in 20,000 years, which would contain 7,304,845 days
>         under this rule.

Simon Cassidy responds:
   This proposal (of Herschel's) would actually make the Gregorian
calendar less accurate. The Vernal equinox is currently drifting
at a rate of about 1.15 calendar hours every 4 centuries (11.5 hours
in 4 millenia). With Herschel's 4-millenia exception kludged onto the
Gregorian rule, the Vernal Equinox would be currently drifting at a
rate of about 1.25 calendar hours in 4 centuries (12.5 hours in 4
millenia), but in the opposite direction!
   The only consequence, of the Herschel 4-millenia-kludge, that might
recommend it to the aims of the Gregorian calendar, is that, the new
error it would introduce, would tend to make the average astronomical
vernal-equinox, at least upon the first application of the 4000-year
exception, move towards the ecclesiastical equinox, rather than away
from it! This is because the average Vernal equinox currently occurs
at about 5p.m. G.M.T. March 20th. (i.e. the correction in 1582 should
have been eleven days, as John Dee insisted, not ten days as implemented
by Pope Gregory and Christopher Clavius).
 
Walter Ziobro continued:
>                 Now the actual tropical year has been measured to
>         365.24219878-6.14*10**-6Te days of ephemeris time at
>         Jan. 1, 1900, where -6.14*10**-6Te represents the fact
>         that the length of the tropical year is slowing down by
>         6.14 days in 1,000,000 Julian centuries (i.e. Te) of
>         36,525 ephemeris days. .....

Simon Cassidy responds:
   Here is the heart of Walter's error! He has assumed (like almost
all calendar scholars to date) that the astronomer's mean tropical year
is the "actual" year length, appropriate for comparing with the average
year of our solar calendar. This is either because he has unquestioningly
swallowed the mischaracterisation by astronomers of their mean tropical
year as "the mean interval between Vernal Equinoxes", or he is deliberately
trying to subvert the "Nicene" intent of the sixteenth century reformers
of the Julian calendar.

Walter Ziobro continued:
>         ....................This means that there are actually
>         7,304,843.97437 days in the 20,000 tropical years from
>         Jan. 1, 1900 A.D., which shows that, even with the proposed
>         4,000 year rule, the Gregorian Calendar is too long by almost
>         one day in 20,000 years.

Simon Cassidy responds:
   Even if we continue with Walter's reasoning (for example, by adopting
the dubious stance that the calendar ought to be regulated by the
astronomer's mean tropical year, rather than the Nicene-mandated
Vernal-Equinox year) then we must correct Walter's faulty mathematics.
If we assume the formula he gives for the length of the astronomer's
mean tropical year (in ephemeris days), then the length of this tropical
year after 20,000 such tropical years (Te = 199.995 Julian centuries)
would be 365.24219878-0.00000614*199.995 which is 365.24097081
ephemeris days.
   Thus the average such year, over the 20,000 year interval from
1900 A.D., would be 365.241584795 ephemeris days. This gives a total
length for that 20,000 years of 7,304,831.7 days (not 7,304,843.97 days
as Walter finds).
   Moreover, these results are in EPHEMERIS days, which are not the same
as calendar days (Universal days) due to the changing rate of the
Earth's rotation. Since the Earth's rotation appears to be slowing,
we can say that these 7,304,831.7 ephemeris days will probably experience
less real days, since more than one ephemeris day will probably be
required (on avergae) for each future actual rotation of the Earth.
But projecting such imponderables more than a millenia or so into the
future is definitely a futile exercise, vitiating the rest of Walter's
attempt to improve the Gregorian average year (continued hereunder):

>                 Now, without any adjustment, the Leap-Week Calendar
>         would contain 7,304,850 days, or 1,043,550 weeks in
>         20,000 years, too long by six days.  I propose that the
>         Leap-Week Calendar drop one additional leap-week in the
>         375th year of the last 400 year cycle of the 20,000 year
>         period, that is, the 19,975th year of that period, resulting
>         in there then being 7,304,843 days, or 1,043,549 weeks in
>         20,000 years. This is now too short by one day, so I further
>         propose that the Leap-Week Calendar continue to drop the
>         additional leap-week in the 19,975th year of the next six
>         20,000 year periods, until the year 139,975 in the seventh
>         20,000 year period is reached, when the leap-week will not
>         be dropped.  Consequently, there would be 51,133,908 days,
>         or 7,304,844 weeks in 140,000 years of the Leap-Week Calendar.
>         This compares to 51,133,907.82060 ephemeris days in the
>         140,000 tropical years from 1900 A.D. to 141,900 A.D.,
>         according to the formula noted previously.
 

Simon Cassidy responds to Walter's general idea of a Leap-Week Calendar:

There already is a Leap Week Calendar, it is the week count of the Int'l
Org. for Standardisation. As for improving its regularity, this issue
was discussed by Chris Carrier and myself earlier (my message entitled
"Re: Calendar Reform, ISO weeks and Lunar calendar." follows hereunder):
 

Chris Carrier wrote:
> However, if we had a calendar which had 52 weeks in a year, and every
> 5 or 6 years had a 53rd week, as in the week-count used by the
> International Organization for Standardization, the religious objection
> to the disruption of the seven-day week would be addressed.

Simon responded:
The week-count of the International Organization for Standardization (herein
referred to henceforth as "ISO") is algorithmically tied (as presently defined)
to the leap-year rule of the Gregorian calendar. Thus, it has a 53rd. week
(or "leap-week") every 5 or 6 or 7 Gregorian years.

The ISO "leap-weeks" follow a 400-year cycle (equivalent to the Gregorian
400-year "dominical letter" cycle). Leap-weeks separated by 7 years
occur once every four-hundred Gregorian years (e.g. the pair of 53-week
ISO-years of 2296 and 2303, Gregorian).

The Julian, Dee-Khayyam and Bonavian-style years have ISO-style leap-weeks
every 5 or 6 years with NO 7-year gaps. The Julian calendar has a 28-year
ISO-style leap-week cycle, the Dee-Khayyam year has a 231-year cycle and
the Bonavian has an 896-year cycle.

The advantage of the Bonavian systems w.r.t the 7-day week is that we don't
have to keep track of entities equivalent to the dominical letters at all,
BUT we do have to keep track of an 896-year cycle.

The Gregorian and Dee-Khayyam years both lead to elaborations of the simple
Julian 28-year week-day (or dominical-letter or "ISO-style leap-week") cycle.

The Gregorian calendar follows a Julian-style 28-year cycle for one or two
centuries at a time with discontinuities at three out of four century
transitions.

The Dee-Khayyam calendar's week-day behaviour can be modelled as a 28-year
Julian-style week-day cycle followed by a 5-year segue into the next 28-year
plus 5-year cycle. After 7 such 33-year periods it begins the complete
week-day cycle again (after 231 years).

An elegant feature of the Dee-Khayyam solar calendar is the way it can
be used to correct the Metonic solunar calendar-cycle with a 231-year
correction-cycle which would be in perfect synchronisation with its
week-day/dominical-letter/ISO-style-leap-week cycle.

The Julian Metonic solunar cycle modelled 235 lunar months as exactly 19
Julian years and was used to regulate the ecclesiastical lunar calendar
and determine the first full moon after March 21st. of each year (March
21st. being the "ecclesiastical Vernal Equinox"). Dionysius Exiguus laid
out this cycle for the determination of Easter at the same time as he
formalised the Anno Domini system (his tables beginning at 532 A.D.).

Taking 235 lunar months as 19 Julian years (of 365.25 days each) resulted
in a long-term error (in following the actual lunar phases) of one day
about every 31 decades (or about 8 days in 25 centuries).
 
When the Vatican reformed the Julian leap-year rule in 1582 they also
implemented a correction to the Julian Metonic solunar cycle. In harmony
with their century-year corrections to the leap-year rule they decided to
also restrict lunar corrections to occur only in century years.

The Gregoran reform decreed that the new "Gregorian" leap-year rule required
correcting the ecclesiastical Metonic cycle backwards by one day in 3 of
every 4 century-years just to keep it "Julian". Then, to correct the error in
the "Julian" Metonic cycle, it decreed a forwards adjustment of one day in
8 of every 25 century years. This combination of two correction cycles results
in 75 backwards steps and 32 forwards steps every 100 centuries. These "steps"
are backwards and forwards through a set of "epact tables", so that for each
new century one must decide which of the two correction cycles applies, or
whether no change is required (because they cancel or neither cycle applies).
 
By discarding the peculiar Gregorian emphasis on century years, the Dee-
Khayyam calendar can correct a Metonic lunar cycle (such as the epact tables
of the Catholic church) by simply smoothing out the correction steps to once
backwards every 231 years (i.e. about 43.3 steps every 100 centuries compared
to the Gregorian's 43 steps aggregate). The resulting calendar-lunation-lengths
are as follows:

A) With "steps" interpreted as "days"
The Gregorian calendar's average lunation-length is    29.530592 calendar days.
The Dee-Khayyam 231-year-regulated lunation-length is  29.530589 calendar days.

B) With "steps" interpreted as "tithis" (thirtieth parts of a lunation)
The Gregorian calendar's average lunation length is    29.530587 calendar days.
The Dee-Khayyam 231-year-regulated lunation length is  29.530583 calendar days.
 

The advantage of a 231-year metonic correction is that it can be expressed in
terms of the week-day cycle of a 33-year solar calendar as follows:

Apply a single step correction to a Metonic solunar cycle whenever Moon day
(Monday) February 29th. is followed (five years later) by Sunday Feb 29th.

--
Dee's Y'rs, Simon Cassidy, 1053 47th.St. Emeryville Ca.94608. ph.510-547-0684.